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Theorem rspcegf 39496
Description: A version of rspcev 3340 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypotheses
Ref Expression
rspcegf.1 𝑥𝜓
rspcegf.2 𝑥𝐴
rspcegf.3 𝑥𝐵
rspcegf.4 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
rspcegf ((𝐴𝐵𝜓) → ∃𝑥𝐵 𝜑)

Proof of Theorem rspcegf
StepHypRef Expression
1 rspcegf.2 . . . 4 𝑥𝐴
2 rspcegf.3 . . . . . 6 𝑥𝐵
31, 2nfel 2806 . . . . 5 𝑥 𝐴𝐵
4 rspcegf.1 . . . . 5 𝑥𝜓
53, 4nfan 1868 . . . 4 𝑥(𝐴𝐵𝜓)
6 eleq1 2718 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
7 rspcegf.4 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜓))
86, 7anbi12d 747 . . . 4 (𝑥 = 𝐴 → ((𝑥𝐵𝜑) ↔ (𝐴𝐵𝜓)))
91, 5, 8spcegf 3320 . . 3 (𝐴𝐵 → ((𝐴𝐵𝜓) → ∃𝑥(𝑥𝐵𝜑)))
109anabsi5 875 . 2 ((𝐴𝐵𝜓) → ∃𝑥(𝑥𝐵𝜑))
11 df-rex 2947 . 2 (∃𝑥𝐵 𝜑 ↔ ∃𝑥(𝑥𝐵𝜑))
1210, 11sylibr 224 1 ((𝐴𝐵𝜓) → ∃𝑥𝐵 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wex 1744  wnf 1748  wcel 2030  wnfc 2780  wrex 2942
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rex 2947  df-v 3233
This theorem is referenced by:  rspcef  39555  stoweidlem46  40581
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